Antilog Calculator

Calculate the inverse logarithm (antilogarithm) for any base

Enter the base and logarithm value to find the original number. Antilog is the inverse operation of logarithm, calculated as base raised to the power of the value.

Examples

Click on any example to load it into the calculator

Common Antilog (Base 10)

common

Standard antilog calculation with base 10

Base: 10

Value: 3

Natural Antilog (Base e)

natural

Natural antilog using Euler's number as base

Base: 2.71828

Value: 2

Binary Antilog (Base 2)

binary

Antilog calculation with base 2 for binary systems

Base: 2

Value: 5

Negative Logarithm Value

negative

Antilog with negative logarithm value

Base: 10

Value: -2

Other Titles
Understanding Antilog Calculator: A Comprehensive Guide
Master the mathematical operation of antilogarithm and its applications in chemistry, physics, and mathematical analysis

What is Antilog? Mathematical Foundation and Core Concepts

  • Antilog represents the inverse operation of logarithm
  • The fundamental relationship: if log_b(x) = y, then antilog_b(y) = x
  • Essential operation in scientific calculations and data analysis
Antilogarithm, commonly abbreviated as 'antilog', is the inverse operation of logarithm. Just as subtraction undoes addition and division undoes multiplication, antilog undoes logarithm to retrieve the original number.
The mathematical relationship is elegantly simple: if logb(x) = y, then antilogb(y) = x. This means that antilog_b(y) = b^y, where b is the base, y is the logarithm value, and the result is the original number x.
Understanding antilog is crucial because it allows us to work backwards from logarithmic scales to linear scales, which is essential in many scientific and engineering applications where data spans multiple orders of magnitude.
Common Types of Antilog:
  • Common Antilog (Base 10): antilog₁₀(y) = 10^y, used in pH calculations, earthquake measurements, and sound intensity
  • Natural Antilog (Base e): antilog_e(y) = e^y, fundamental in exponential growth, radioactive decay, and financial modeling
  • Binary Antilog (Base 2): antilog₂(y) = 2^y, essential in computer science and information theory

Fundamental Antilog Examples

  • antilog₁₀(3) = 10³ = 1000 - if log₁₀(1000) = 3, then antilog₁₀(3) = 1000
  • antilog_e(2) = e² ≈ 7.389 - natural antilog for exponential calculations
  • antilog₂(5) = 2⁵ = 32 - binary antilog for computer applications
  • antilog₁₀(-2) = 10⁻² = 0.01 - antilog with negative logarithm value

Step-by-Step Guide to Using the Antilog Calculator

  • Master the input format for base and logarithm values
  • Understand different base types and their applications
  • Interpret results and verify calculations effectively
Our antilog calculator provides an intuitive interface for computing antilogarithms with professional-grade accuracy across all common bases.
Input Guidelines:
  • Base (b): Enter any positive number except 1. Common values include 10 (common log), 2.71828 or 'e' (natural log), and 2 (binary log).
  • Logarithm Value (y): Enter any real number including decimals and negative values. This is the result of the original logarithm operation.
  • Decimal Precision: The calculator handles high-precision decimals for accurate scientific calculations.
Calculation Process:
The calculator performs the computation x = b^y, where b is your base and y is your logarithm value. The result x is the original number that, when subjected to log_b, would yield y.
Verification Methods:
To verify your result, you can check that log_b(result) equals your original logarithm value. This reverse calculation confirms the accuracy of your antilog computation.
For common calculations, you can use mental math: antilog₁₀(2) = 100, antilog₁₀(3) = 1000, antilog₁₀(0) = 1, providing quick reference points.

Practical Calculation Examples

  • Input: Base = 10, Value = 2 → Output: 100 (because 10² = 100)
  • Input: Base = e, Value = 1 → Output: ≈2.718 (because e¹ = e)
  • Input: Base = 2, Value = 8 → Output: 256 (because 2⁸ = 256)
  • Input: Base = 10, Value = -1 → Output: 0.1 (because 10⁻¹ = 0.1)

Real-World Applications of Antilog in Science and Engineering

  • Chemistry: pH calculations and concentration determinations
  • Seismology: Earthquake magnitude and energy calculations
  • Acoustics: Sound intensity and decibel conversions
  • Finance: Compound interest and exponential growth modeling
Antilog calculations are fundamental to numerous scientific and engineering applications where logarithmic scales are used to handle data spanning multiple orders of magnitude.
Chemistry - pH and Concentration:
pH is defined as -log₁₀[H⁺], where [H⁺] is hydrogen ion concentration. To find concentration from pH: [H⁺] = antilog₁₀(-pH) = 10⁻ᵖᴴ.
This calculation is essential in analytical chemistry, environmental monitoring, and pharmaceutical development where precise concentration measurements are critical.
Seismology - Earthquake Analysis:
The Richter scale uses logarithmic measurement: M = log₁₀(A/A₀). To compare earthquake amplitudes: A = A₀ × antilog₁₀(M) = A₀ × 10ᴹ.
This allows seismologists to quantify the relative strength of earthquakes and assess potential damage levels.
Acoustics - Sound Measurement:
Sound intensity in decibels: L = 10 × log₁₀(I/I₀). To find actual intensity: I = I₀ × antilog₁₀(L/10) = I₀ × 10^(L/10).
This is crucial in audio engineering, noise control, and hearing protection where precise intensity measurements guide design decisions.
Financial Mathematics:
Compound interest with continuous compounding: A = P × e^(rt). When given logarithmic growth rates, antilog calculations determine final amounts and investment projections.

Scientific Application Examples

  • pH 3 solution: [H⁺] = 10⁻³ = 0.001 mol/L (acidic solution)
  • Magnitude 7 earthquake: 10 times stronger amplitude than magnitude 6
  • 80 dB sound: 100 times more intense than 60 dB sound
  • Investment growth: If ln(A/P) = 0.693, then A = P × e^0.693 = 2P (doubled)

Common Misconceptions and Calculation Pitfalls

  • Antilog is not the reciprocal of logarithm (1/log)
  • Base specification is crucial for accurate results
  • Understanding domain and range limitations
Misconception 1: Antilog as Reciprocal
A frequent error is confusing antilog with the reciprocal: antilogb(y) ≠ 1/logb(y). Antilog is the inverse function (exponentiation), not the multiplicative inverse.
Correct understanding: If log₁₀(100) = 2, then antilog₁₀(2) = 100, not 1/2 = 0.5.
Misconception 2: Base Ambiguity
The term 'antilog' without base specification is meaningless. Always identify whether you're dealing with common antilog (base 10), natural antilog (base e), or another base.
Context clues: 'log' typically means base 10, 'ln' means base e, and 'log₂' means base 2. The antilog must use the same base.
Misconception 3: Domain Restrictions
Remember that logarithm bases must be positive and not equal to 1. While logarithm values can be any real number, the base restriction is absolute.
Calculation Best Practices:
Always verify your answer by computing log_b(result) and checking if it equals your original logarithm value. This reverse calculation catches input errors and computational mistakes.
For scientific applications, pay attention to significant figures and rounding. The precision of your antilog result should match the precision of your input data.

Common Errors and Corrections

  • Correct: antilog₁₀(2) = 10² = 100
  • Incorrect: antilog₁₀(2) ≠ 1/log₁₀(2) ≠ 1/2
  • Base matters: antilog₁₀(3) = 1000, but antilog₂(3) = 8
  • Verification: log₁₀(100) = 2 confirms antilog₁₀(2) = 100

Mathematical Derivation and Advanced Examples

  • Formal definition of inverse logarithmic functions
  • Properties and rules of antilogarithm operations
  • Complex calculations and multi-step problems
Formal Mathematical Definition
If f(x) = logb(x), then the inverse function f⁻¹(y) = antilogb(y) = b^y. This establishes antilog as the true mathematical inverse of logarithm.
Domain and Range: For antilog_b(y), the domain is all real numbers y, and the range is all positive real numbers x > 0.
Properties of Antilog Operations
  • Identity Property: antilogb(logb(x)) = x for all x > 0
  • Addition Rule: antilogb(y₁ + y₂) = antilogb(y₁) × antilog_b(y₂) = b^(y₁+y₂) = b^y₁ × b^y₂
  • Subtraction Rule: antilogb(y₁ - y₂) = antilogb(y₁) ÷ antilog_b(y₂) = b^(y₁-y₂) = b^y₁ ÷ b^y₂
  • Multiplication Rule: antilogb(n × y) = [antilogb(y)]ⁿ = (b^y)ⁿ = b^(ny)
Change of Base Formula for Antilog
To convert between different bases: antilogb(y) = antilogc(y × logc(b)) = c^(y × logc(b))
This formula enables calculation of any base antilog using a calculator that only supports specific bases (typically base 10 or e).
Advanced Problem-Solving Techniques
For complex problems involving multiple logarithmic operations, break down the calculation into steps, apply antilog properties systematically, and verify each intermediate result.
When dealing with scientific notation results, express answers in appropriate form: antilog₁₀(4.5) = 10^4.5 ≈ 3.16 × 10⁴ = 31,623.

Advanced Mathematical Examples

  • Property verification: antilog₁₀(log₁₀(50)) = 50
  • Addition rule: antilog₁₀(2 + 3) = antilog₁₀(2) × antilog₁₀(3) = 100 × 1000 = 100,000
  • Base conversion: antilog₂(5) = antilog₁₀(5 × log₁₀(2)) = 10^(5×0.301) ≈ 32
  • Scientific notation: antilog₁₀(6.5) = 10^6.5 ≈ 3.16 × 10⁶ = 3,162,278